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In QM of finitely many degrees of freedom it is well known that due to the Stone-Von Neumann theorem, the CCR

$$[Q_i,P_j]=i \delta_{ij} $$

leads to a unique representation up to unitary equivalence, on which $P_j$ acts as the derivative

$$P_j\mapsto -i\partial_j.$$

Now, in Weinberg's QFT book volume 1, chapter 9, he considers a general quantum mechanical system with coordinates $Q_a$ and momenta $P_b$ satisfying

$$[Q_a,P_b]=i\delta_{ab},\tag{9.1.1}$$

with $a,b$ allowed to have a continuous part. So that in quantum field theory we would have for instante $a\mapsto \mathbf{x}$ so that we have coordinates $\phi(\mathbf{x})$ and momenta $\pi(\mathbf{x})$ and then $\delta_{ab}$ turns into $\delta(\mathbf{x}-\mathbf{y})$.

Weinberg then considers the eigenstate basis of both operators

$$Q_a|q\rangle = q_a |q\rangle,\tag{9.1.6}$$ $$ P_b|p\rangle = p_b |p\rangle,\tag{9.1.9}$$

where $q,p$ stands for the whole collection of $q_a$ and $p_b$.

He then claims that

$$\langle q | p\rangle = \prod_a \dfrac{1}{\sqrt{2\pi}} \exp (i p_a q_a)\tag{9.1.12}$$

with a footnote on page 379 saying that it follows from the CCR that $P_b$ acts as $-i\partial/\partial q_b$ on wave functions and thus this formula follows.

Well there are quite a few objections here:

  1. If the labels $a,b$ have infinite values, as it happens in QFT, we have a system with infinitely many degrees of freedom to which the Stone-Von Neumann theorem doesn't apply. There are infinitely many inequivalent representations of the CCR and we can't conclude that $P_b$ acts as a derivative as suggested by the author.

  2. More alarming would be that if the labels have infinite values, the expression for $\langle q | p\rangle$ have $(2\pi)^{-1/2}$ multiplying itself infinite times, which isn't well defined.

So how to make sense of these issues? What am I missing here?

Qmechanic
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Gold
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  • Weinberg is writing infinite products that need to be regularized to make mathematical sense. 2. Concerning the overlap formula (9.1.12) in finite dimensions, see this Phys.SE post.
    • – Qmechanic May 19 '18 at 18:50
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    I don’t think Weinberg is meant to be a formal QFT book. Do you have anything against putting the theory on a lattice in a box? – knzhou May 19 '18 at 19:00
  • So what you mean is: we should discretize space, to get a theory on a lattice with spacing $\epsilon$. Then we have finitely many degrees of freedom (so that SvN applies) since we get finite values of $\mathbf{x}$ and hence the label $a$ becomes discrete and finite. Thus we should understand $\langle q |p \rangle$ as the limit of the expression obtained in that setting as the lattice spacing $\epsilon \to 0$? – Gold May 19 '18 at 19:09
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    If you’re willing to believe that limit exists, sure! In the physical world, it doesn’t matter, so you don’t really have to take the limit if you don’t want to. – knzhou May 19 '18 at 19:38